For log interpretation for shaly sand reservoirs, there are several models available. Commonly used models include the Waxman and Smits (W-S), 1968 model, and dual water (D-W) model of Clavier et al., 1984. See, Waxman and Smits, Electrical conductivities in oil-bearing shaly sands, SPEJ 8(2), 107-122, (1968); and Clavier, et al, The theoretical and experimental bases for the dual water model for the interpretation of shaly sands, SPE 6859, 1977 ATCE, SPEJ April (1984). Although these models have successes in the interpretation of electric-log responses of shaly sand homogeneous reservoir rocks, the models are not explicit in their predictions of electrical conductivity with respect to rock structure, spatial fluid distribution in the pore space, wettability, or clay mineral distribution. See, Devarajan, S., Toumelin, E., Torres-Verdín, C., Thomas, E. C., “Pore-scale analysis of the Waxman-Smits shaly sand conductivity model”, SPWLA, Jun. 4-7 (2006). The models rely on information about clay cation exchange capacity (CEC) and formation water salinity (Rw) as demonstrated infra.
Archie, G. E., the Electrical Resistivity Log as an Aid in Determining Some Reservoir Characteristics, Trans. of AIME 146 (1), (1942) discusses the fundamental empirical correlation for interpreting conductivity measurements:σT=φmSwnσw Where σT is formation true resistivity, σw is formation water resistivity, Sw is water saturation, n is saturation exponent, φ is reservoir total porosity, and m is cementation exponent.
When clay minerals are present, Waxman-Smits empirical model can be applied and it is characterized by the following equation:
      σ    t    =            φ      m        ⁢                  S        w        n            ⁡              (                              σ            w                    +                                    BQ              v                                      S              w                                      )            Where, m* and n* are Archie cementation and saturation exponents for shaly sands applied to total pore volume. B is specific cation conductance in (ohm−1)/(meq/ml), Qv is the cation exchange capacity (CEC) per unit pore volume:
      Q    v    =            ρ      g        ⁢    CEC    ⁢                  1        -        φ            φ      Where CEC is in meq/gram of dry rock, ρg is rock grain density in g/cc, and φ is total porosity. In clean zones (no clay), CEC=0, thus Qv=0, m*=m, n*=n, and the W-S model becomes Archie model.
The D-W model has been developed based on the double layer effect close to the grain surface, the D-W equation:
      σ    t    =            φ              m        0              ⁢                  S        w                  n          0                    ⁡              [                              σ            wF                    +                                                    S                wB                                            S                w                                      ⁢                          (                                                σ                  wB                                +                                  σ                  wF                                            )                                      ]            Where σwB is clay bound water resistivity, σwF is free formation water resistivity, SwB is clay bound water saturation with respect to total pore volume, and can be estimated using the HSK model proposed by Hill, H. J., Shirley, O. J., and Klein, G. E, Bound Water in Shaly Sands—Its Relation to Qv and Other Formation Properties. The Log Analyst 20 (3): 3 (1979):
      S    wB    =            Q      v        ⁡          (                                    a            1                                              C              NaCl                                      +                  a          2                    )      Where a1 and a2 are constants and CNaCl is NaCl concentration in equivalent/liter. In clean zones (no clay), SwB=0, m0=m, n0=n, the D-W model becomes the Archie model.
CEC is measured in the laboratory by potentiometric titration methods. See, Meier, L. P., and G. Kahr, Determination of the cation exchange capacity (CEC) of clay minerals using the complexes of copper (II) ion with triethylenetetramine and tetraethylenepentamine, Clays Clay Miner, 47(3), 386-388 (1999). Uncertainties associated with this laboratory measurement are many, including how representative of the laboratory sample analyzed to downhole conditions (clays can be sensitive to environment changes) and details of the laboratory sample preparation and analysis such as the degree to which the clay mineral geometry is altered by the disaggregation of the core sample, which can be enhanced by grinding to grain size particles. See, Huff, G. F., A Correction for the Effect of Comminution on the Cation Exchange Capacity of Clay-Poor Sandstones, SPE Form Eval 2 (3): 338-344, SPE-14877 (1987).
Formation water salinity or formation water resistivity Rw can be obtained by water analysis in laboratory. See, Ma, S., Hajari, A., Berberian, G. & Ramamoorthy, R: “Cased-Hole Reservoir Saturation Monitoring in Mixed Salinity Environments—A New Integrated Approach,” SPE 92426 MOES (2005). Without a robust continuous in-situ measurement, formation water salinity is often assumed to be constant within the hydrocarbon column, and usually there is little data regarding Rw other than from formation sampling. In several cases in which the Rw distribution has been studied in depth, it was found to vary in systematic ways within the hydrocarbon column. See, McCoy, D. and Fisher, T. E., Water-Salinity Variations in the Ivishak and Sag River Reservoirs at Prudhoe Bay, SPE Res Eng 12 (1): 37-44, SPE-28577 (1997); Rathmell, J. J., Bloys, J. B., Bulling, T. P. et al., Low Invasion, Synthetic Oil-Base Mud Coring in the Yacheng 13-1 Gas Reservoir for Gas-in-Place Calculation, Presented at the International Meeting on Petroleum Engineering, Beijing, China, 14-17 November SPE-29985 (1995); and Rathmell, J., Atkins, L. K., and Kralik, J. G., Application of Low Invasion Coring and Outcrop Studies to Reservoir Development Planning for the Villano Field, Presented at the Latin American and Caribbean Petroleum Engineering Conference, Caracas, Venezuela, 21-23 April, SPE-53718 (1999).
Efforts have been made recently to derive this salinity information from special wireline logs. See, Ma, S., Pfutzer, H., Hajari, A., Musharfi, N., Saldungaray, P. & Azam, H: “Resolving Mixed Salinity Challenge with a Methodology Developed from Pulsed Neutron Capture Gamma Ray Spectral Measurements,” SPE 170608, SPE ATCE, Amsterdam, Oct. 27-29 (2014).